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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The rectangle method (also called the midpoint rule) is the simplest method in Mathematics used to compute an approximation of a definite integral."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's check this method for the next function: $$f(x) = ({e^x / 2})*(cos(x)-sin(x))$$ with $\\varepsilon = 0.001$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import math \n",
    "import numpy as np\n",
    "\n",
    "def integration(a,b,n):\n",
    "    h = (b-a)/n\n",
    "    r = f(a) + f(b)\n",
    "    i = 1\n",
    "    while i < n:\n",
    "        x = a + i*h\n",
    "        r = r + 4 * f(x)\n",
    "        i = i + 1\n",
    "        x = a + i * h\n",
    "        r = r +2*f(x)\n",
    "        i = i + 1\n",
    "    r = r * h / 3\n",
    "    print(\"Result: \", r) \n",
    "\n",
    "def rectangles(a,b,n):\n",
    "    \n",
    "    z = (b-a)/n\n",
    "    i = a\n",
    "    s1=0\n",
    "    s2=0\n",
    "    while i<b:\n",
    "        \n",
    "        s1=s1+f(i)*z\n",
    "        i=i+z\n",
    "    i=a \n",
    "    while i<b:\n",
    "        i=i+z\n",
    "        s2=s2+f(i)*z\n",
    "\n",
    "    print('Result of formula of the left rectangles: ',s1)\n",
    "    print('Result of formula of the left rectangles: ',s2)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Some input data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Result:  -10.297317477276613\n",
      "Result of formula of the left rectangles:  -7.576924395545134\n",
      "Result of formula of the left rectangles:  -9.192576890365931\n"
     ]
    }
   ],
   "source": [
    "def f(x):\n",
    "    return  (math.e**x / 2)*(math.cos(x)-math.sin(x))\n",
    "\n",
    "n = 4  \n",
    "a = 2.\n",
    "b = 3.\n",
    "Si = []\n",
    "\n",
    "integration(a,b,n)\n",
    "rectangles(a,b,n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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